Localized Energy Estimates for Wave Equations on 5D Myers-Perry Space-Times (2014)

Undergraduates: Shreyas Tikare, Parul Laul, Jason Metcalfe, Mihai Tohaneanu

Faculty Advisor: Jason Metcalfe
Department: Mathematics

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A robust measure of decay and dispersion for the wave equation is provided by the localized energy estimates, which have been essential in proving, e.g. the Strichartz estimates on black hole backgrounds. We study localized energy estimates for the wave equation on 5D Myers-Perry space-times, which represent a family of rotating, axially symmetric, asymptotically flat black holes with spherical horizon topology and generalize the well-known Kerr space-times to higher dimensions. The Myers-Perry family is parameterized by two angular momentum parameters, which we assume to be sufficiently small relative to the mass of the black hole. This investigation is motivated by the nonlinear stability problem for the Kerr family of black holes, which may be easier to understand in higher dimensions. Typically, the localized energy estimates are proved by commuting the wave operator with a suitable first-order differential operator and integrating by parts. However, the underlying black hole geometry introduces a number of difficulties related to the trapping phenomenon, which is a known obstruction to dispersion and necessitates a loss in decay. On the Myers-Perry space-time, the nature of the trapped set is quite complicated in the sense that a single differential multiplier is insufficient to prove the desired result. As in the work of Tataru-Tohaneanu, we instead commute with an appropriate pseudodifferential operator to generate a positive commutator near the trapped set.